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A Probabilistic Proof Of An Identity Related To The Stirling Number Of The First Kind

In: Recent Advances In Stochastic Operations Research II

Author

Listed:
  • MITSUSHI TAMAKI

    (Faculty of Business Administration, Aichi University, Miyoshi, Aichi, Japan)

Abstract

The basic assumption of the infinite formulation of the secretary problem, originally studied by Gianini and Samuels, is that, if Uj, j = 1, 2,…, is defined as the arrival time of the jth best from an infinite sequence of rankable items, then U1, U2,…, are i.i.d., uniform on the unit interval (0, 1). An item is referred to as a record if it is relatively best. It can be shown that a well known identity related to the Stirling number of the first kind, as given in Eq.(3) in this note, is just the identity obtained through the derivation of the probability mass function of the number of records that appear on time interval (s, t), 0 < s < t < 1, in two ways in the infinite formulation.

Suggested Citation

  • Mitsushi Tamaki, 2009. "A Probabilistic Proof Of An Identity Related To The Stirling Number Of The First Kind," World Scientific Book Chapters, in: Tadashi Dohi & Shunji Osaki & Katsushige Sawaki (ed.), Recent Advances In Stochastic Operations Research II, chapter 1, pages 3-9, World Scientific Publishing Co. Pte. Ltd..
    • Handle: RePEc:wsi:wschap:9789812791672_0001
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